CounterexampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Mass can escape to infinity under pointwise convergence

Statement refuted

Pointwise convergence of nonnegative integrable functions forces convergence of their integrals.

Facts & Assumptions

Given: The travelling-mass sequence $f_{n} : = χ_{[n, n + 1]}$ on $R$ .

[L1]

Fatou's lemma only compares $\int lim inf f_{n}$ with $lim inf \int f_{n}$ (Fatou's lemma).

Counterexample

technique · direct

1.1

For each fixed $x \in R$ , the value $f_{n} (x)$ is eventually $0$ , so $f_{n} (x) \to 0$ .

given

2.1

Yet $\int f_{n} d λ = 1$ for every $n$ . Thus the mass has escaped to infinity instead of disappearing, and the Statement is false. This is the strict case already permitted by [L1].

step 1.1L1algebra∎

Depends on

Fatou's lemma

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John K. Hunter, Measure Theory Notes, Example 4.20 (standard reference, not scraped)